Proposition 1

Let $t_0$ be a point of $I$ and $x_0$ a point of $H$; for a continuous function $u$ to be a solution of the equation (1) on $I$ and take the value $x_0$ at the point $t_0$, it is necessary and sufficient that it satisfies the relation

for every $t \in I$.

Indeed, by lemma 1, if $u$ is a solution of (1) on $I$, then $f(t, u(t))$ is regulated, so the right-hand side of (6) is defined and equal to $u(t)$ for every $t \in I$. Conversely, if $u$ is a continuous function that satisfies (6) then $f(t, u(t))$ is regulated, by lemma 1, so $u$ has derivative equal to $f(t, u(t))$ except at the points of a countable subset of $I$.